Download Public and Private Production in a Two-Sector Economy

Public and Private Production in a Two-Sector Economy
Mihaela Iulia Pintea
Florida International University
Stephen Turnovsky
University of Washington
Abstract
We develop a two-sector “non-scale” production model in which there are two types of firms, conventional profitmaximizing private firms, and what we call “public firms”, whose objective is to produce a specified quantity of
government investment goods – determined by government policy – at minimum cost. Furthermore, the production
functions of the two sectors need not in general coincide. Using this two-sector production set-up we characterize the
equilibrium dynamics, and analyze a variety of fiscal disturbances. Because of the complexity of the model our analysis
is carried out using simulations of a calibrated economy. We find that the effects of tax policies are remarkably robust
with respect to the relative capital intensities of the two productive sectors. In contrast, the effects of government
investment are much more sensitive to this aspect. One conclusion of this is that one can continue to employ the onesector Ramsey model to analyze tax policy in the presence of public capital without seriously jeopardizing the analysis.
But one has to be more careful in analyzing the impact of public investment itself.
1
1.
Introduction
During recent years an increasing volume of research has focused on the role of government
capital as a critical productive input. Without exception, this has been carried out using a onesector model of production, in which public capital enters the aggregate production function
together with private capital and labor; see e.g. Arrow and Kurz (1970), Baxter and King (1993),
Futagami, Murata, and Shibata (1993), Fisher and Turnovsky (1998), Turnovsky (2003). This
implicitly assumes that all production occurs in the private sector. The government then enters the
private market and makes its purchases, in competition with private agents, using the resources it
generates from tax revenues and from borrowing.
While this is a reasonable description and provides a good working model, in many instances
governments may be better viewed as effectively conducting their own productive operations.
Consider the following stylized description. A government passes legislation to invest in an airport
facility, say. It sets out precise specifications of the project, which it then puts out for bids to
private contractors who will hire labor and employ private capital to carry out the project. But
instead of being free to hire productive factors to maximize his profit -- and thereby determine his
output level (the size of the airport) endogenously -- the contractor is constrained (to win the
contract) to construct the project, as specified by the government, most efficiently. Moreover, there
is no reason to assume that the technology employed in the public project need be the same as that
in the private sector. Indeed, one can plausibly argue that government investment projects that
involve the nation’s infrastructure may well be more capital intensive (in private capital) than is the
average technology employed in the private sector.
In this paper we develop a two sector model in which private output is produced in one sector
by profit-maximizing firms. The production function in that sector depends upon the stocks of both
private and public capital, as well as upon endogenously supplied labor. Public capital introduces a
positive externality in production, so that the complete production function is one of overall
increasing returns to scale in these three productive factors. Government capital is produced in a
second sector by what we shall term “public firms”, hiring labor and private capital, with public
capital also providing an externality. In hiring their productive factors these firms compete with the
2
private sector and therefore need to pay competitive factor returns. The objective of the public firm
is to provide the new public capital, as determined by government policy, at minimum cost. Thus
in contrast to the usual model in which the government uses its resources to finance its direct
purchases of new investment, the government uses its resources to purchase the services of the
productive factors that it employs. These resources are obtained by taxing capital income, labor
income, and consumption, or by imposing non-distortionary lump-sum taxation.
To our knowledge, there are few other papers that disaggregate the production side of the
economy in this way. Schmitz (2001) argues that in many economies, particularly Egypt, and to a
lesser degree India and Turkey, the vast majority of investment goods are produced by the
government, showing that this led to a major reduction in labor productivity. He also extends the
model to allow for both public and private production of the economy’s investment good.1
The model we employ is a “non-scale” growth model of the type introduced by Jones (1995a,
1995b), Segerstrom (1998), and others. This can be viewed as an extension of the neoclassical
model, generalized to allow for non-constant returns to scale.2 Like the neoclassical model, this has
the property that the long-run equilibrium growth rate is determined by the interaction between the
population growth rate and technological production parameters and is independent of government
policy parameters. Also like the neoclassical growth model it has the advantage of being consistent
with balanced growth under quite general production structures.
The fact that the long-run growth rate is independent of fiscal instruments does not mean that
fiscal policy is irrelevant for long-run economic performance. On the contrary, fiscal policy has
important effects on the levels of key economic variables, such as the per capita stock of capital and
output. Moreover, the non-scale model typically yields slow asymptotic speeds of convergence,
consistent with the empirical evidence of 2-3% per annum; see Eicher and Turnovsky (1999).3 This
1
Arrow and Kurz (1970, Chapter 4) discuss a two sector model, referring to the private and public sector, but they do not
disaggregate the production process in the way being proposed here.
2
Interest in this model has grown as a result of increasing concerns about the endogenous growth model, which has
centered around three objections: (i) Its frequent association with “scale effects”, meaning that the equilibrium growth
rate is tied to the size of the economy, (ii) the weak empirical support for some of its policy implications, (iii) the knifeedge conditions on both preferences and technology that are required to generate ongoing growth. These objections, as
well as some critiques of the non-scale model, are reviewed in more detail by Turnovsky (2000).
3
This benchmark was established in early work by Barro (1991), Mankiw, Romer, and Weil (1992). Subsequent studies
suggest that the convergence rates are more variable and sensitive to time periods and the set of countries than originally
suggested and a wider range of estimates have been obtained. For example, Islam (1995) estimates the rate of
3
implies that policy changes can affect growth rates for sustained periods of time so that their
impacts during the transition from one equilibrium to another will eventually accumulate to
potentially large influences on steady-state levels. Thus, although the economy grows at the same
rate across steady states, the corresponding bases upon which the growth rates compound may be
substantially different. These considerations suggest that attention should be directed to analyzing
the impact of fiscal policy on the transitional dynamics and this has been the focus of a number of
previous studies; see e.g. Auerbach and Kotlikoff (1987), Baxter and King (1993), King and Rebelo
(1993), Devereux and Love (1994), Turnovsky and Chatterjee (2002), Turnovsky (2003).
Our objective is to analyze the effects of government investment and changes in the
alternative tax rates in this two-sector productive economy. We set out the dynamic equilibrium of
this economy and show how the stable adjustment is characterized by a two dimensional locus in
terms of the two stationary variables, referred to as “scale-adjusted” per capita stocks of private and
public capital.4 Because of its complexity it is necessary to analyze the equilibrium numerically,
and to do this we calibrate the model to a benchmark economy and assess the numerical effects of
various policy shocks relative to this benchmark. Different fiscal shocks have different impacts on
government revenues, and to preserve comparability the shocks must be normalized in terms of
their impacts on the government deficit. This can be done either with respect to their short-run
effects or their intertemporal effects. We focus on the latter. Both the transitional adjustments and
the eventual long-run equilibrium responses are considered. Particular attention is devoted to the
welfare of the representative agent, both the time profile of instantaneous utility and intertemporal
welfare, as represented by the present value of the accumulated benefits.
The major focus of the paper is to highlight the intertemporal dimensions of fiscal policy and
the tradeoffs these involve for economic performance, especially growth and welfare, paying
convergence to be 4.7% for non-oil countries and 9.7% for OECD economies. Evans (1997) obtains estimates of the
convergence rate of around 6% per annum.
4
The fact that the transitional paths are two-dimensional introduces important flexibility to the dynamics. Convergence
speeds vary over time and across variables, allowing different variables to follow different transitional paths; see Eicher
and Turnovsky (1999). This characteristic is relevant to the empirical evidence of Bernard and Jones (1996), who find
that while growth rates of output among OECD countries converge, the growth rates of manufacturing technologies
exhibit markedly different time profiles. This contrasts with rigid dynamics imposed by the standard one-sector
neoclassical growth model, or the familiar two-sector endogenous growth model.
4
particular attention to the role of the more general production structure. Overall, we find that the
effects of pure tax policies are rather insensitive to the differences in the production structures
between the private and public sectors. Thus, one general conclusion to be drawn is that the
conventional one-sector model is a perfectly adequate vehicle for analyzing the aggregate effects of
pure tax policies.
In contrast, the effects of government investment, irrespective of how they are financed, are
more sensitive to the comparative production structures of the two sectors. The strongest effect is
on the allocation of the private productive factors across the two sectors, but in general normalized
government investment will be significantly more expansionary as the capital intensity of the public
sector increases. These translate to somewhat larger aggregate welfare effects.
Calibrating the model to approximate the US, we find that public investment involves a
dramatic intertemporal tradeoff. Consumption and welfare decline in the short run, as resources are
attracted toward government investment, but improve sharply over time as the investment bears
fruit and productivity is enhanced. For example, doubling the rate of public investment from 4% to
8% involves short-run welfare losses of between 2.5% - 3.2% but intertemporal welfare gains of
between 7.7% - 10.9%, depending upon the relative sectoral capital intensities.
A similar tradeoff exists for the tax on capital income. The normalized reduction in this tax
leads to immediate welfare losses of around 2.9%, as resources are diverted away from
consumption and leisure toward private capital accumulation and labor. But in the long run, the
benefits of more capital generate intertemporal welfare gains of just over 2%, with some minor
variation across the different technologies. The equivalent reductions in the tax on labor income
and consumption have milder effects over time. One unexpected result is that the intertemporal
benefits of the normalized reduction in the tax on labor income exceed those of the tax on capital
income. This is because the normalization is taken with respect to the government’s intertemporal
deficit and the tax on capital has substantial growth effects. If instead, we normalize the tax cuts so
as to have the same impact on the current deficit, this requires a relatively greater cut in the capital
income tax so that its intertemporal effects dominate those of the equivalent labor income tax.
The remainder of the paper proceeds as follows. Section 2 sets out the structure of the
5
model, while its equilibrium dynamics are characterized in Section 3. Section 4 outlines the steadystate equilibrium in the centrally planned economy, which serves as a natural benchmark. Section 5
calibrates the model and considers the numerical effects of a number of policy changes. Section 6
concludes, while technical details of the solution are provided in the Appendix.
2.
The Model
We begin by outlining the structure of the economy.
2.1
Private Firms
There are M identical private firms indexed by j. We shall assume that the representative
private firm produces private output Yj in accordance with the Cobb-Douglas production function
Y j = αY ( K j )b [ L j ]1−b K Gσ
0 < b < 1, σ > 0
(1a)
where K j denotes the capital employed by the firm, L j denotes the labor employed by the firm, and
K G denotes the aggregate stock of public capital. This production function is constant returns to
scale in the two private factors, while public capital, provides a positive externality, and behaves as a
pure public good, not subject to congestion.5 Summing over the M firms yields the aggregate
production function describing output in the private sector
Y = αY (MK j ) b [ML j ]1− b KGσ
(1b)
where Y = MY j denotes aggregate private output.
We assume that the representative private firm maximizes profit, so that the equilibrium rates
of return in the private sector satisfy
r= b
∂Yj
∂K j
=b
Yj
Kj
=b
Y
MK j
;
5
(2a)
This assumption is a polar one, since almost all public services are subject to some congestion. Fisher and Turnovsky
(1998) develop a simple one-sector non-scale model in which productive public capital is subject to congestion. It would
be straightforward to apply their parameterization of congestion to the present two-sector economy and is a direction in
which the present model could be fruitfully extended.
6
w=
∂Y j
∂L j
= (1− b)
Yj
Lj
= (1− b)
Y
ML j
(2b)
where r = gross return to capital, w = (before tax) wage rate.
2.2
Public Firm
We shall assume that there is one public firm, indexed by P, that hires labor and private
capital, paying the market wage and return to capital. The output of this public firm is described by
the production function
J = α J [K P ] [LP )]
d
1− d
η
KG
(3)
which like (1b) is of the Cobb-Douglas form. However, the different exponents allow for potentially
different factor productivities, and therefore equilibrium factor intensities, in the public sector from
those in the private sector. Intuitively, we are assuming that the production of the diverse range of
outputs produced in the private sector (consumption goods as well as private capital) involves a
different technology than does the production of infrastructure produced in the public sector. In this
respect, we shall permit the production function of the public sector to be either more or less capital
intensive than is the representative production function in the private sector.
The public firm’s optimization problem is to produce a specified output of public capital, J,
chosen by the public policy maker, in accordance with the production function (3), at minimum cost.
Formally, this is expressed by
min rK P + wLP + λ  J − α J [ K P ]d [ LP )]1− d K Gη 
(4)
where λ measures the marginal cost of a unit of public investment. Performing the optimization
yields the cost minimizing conditions
r= λ
∂J
J
= λd
∂KP
KP
w=λ
(5a)
∂J
J
= λ(1− d)
∂L P
LP
(5b)
7
Note that the extent λ deviates from unity reflects the extent to which the public firm deviates from
profit maximization in its productive activities. Given J, public capital accumulates at the rate
K& G = J − δ G K G
(6)
where δ G denotes the rate of depreciation of public capital.
2.3
Representative Consumer
The representative consumer is endowed with one unit of time that can be allocated to
leisure, l, leaving (1− l) available for work, ? of which is to employment in the private sector and
(1 − θ ) to employment in the public sector. There are N agents, all identical, and thus we can drop
the subscript on labor. Agent i also owns K i units of private capital, which he rents out at the rental
rate, r, a fraction φ being allocated to the private sector, and 1 − φ to the public sector.
The representative agent’s welfare is specified by the intertemporal isoelastic utility function:
max ∫
∞
0
1 γ ργ − β t
Ci l e dt ;
γ
ρ >0; − ∞ < γ ≤ 1; 1 > γ (1 + ρ )
(7a)
where Ci denotes the agent’s consumption, at time t, 1 (1 − γ ) equals the intertemporal elasticity of
substitution, and ? measures the impact of leisure on the agent’s welfare. The remaining constraints
on the coefficients are imposed to ensure that the utility function is concave in the quantities Ci and l.
The agent’s objective is to maximize (7a) subject to his accumulation equation,
K& i = [(1 − τ k )r − n − δ K ]K i + (1 − τ w )(1 − l ) w − (1 + τ c )Ci − Ti
(7b)
where τ k =tax on capital income, τ w =tax on wage income, τ c =consumption tax, Ti=T/N=agent’s
share of lump sum taxes (transfers). Equation (7b) assumes that private capital depreciates at the rate
δ K , so that with the growing population, the net after tax private return to capital is
(1 − τ k )r − n − δ K .
Performing the optimization yields:
Ciγ −1liργ = µi (1 + τ c )
(8a)
8
ρ Ciγ liρ (γ −1) = µi (1 − τ w ) w
(1 − τ K )r − n − δ K =β −
(8b)
µ&
µ
(8c)
Equation (8a) equates the marginal utility of consumption to the individual’s tax adjusted shadow
value of wealth, µi, while (8b) equates the marginal utility of leisure to its opportunity cost, the after
tax real wage, valued at the shadow value of wealth. The third equation is the standard KeynesRamsey consumption rule, equating the rate of return on consumption to the after-tax rate of return
on capital. Finally, in order to ensure that the agent’s intertemporal budget constraint is met, the
following transversality condition must be imposed:
lim µi K i e − β t = 0
(8d)
t →∞
2.4
Aggregate Relationships
Aggregating over the firms and households, full employment in the labor market and private
capital market is described by
Nθ (1 − l ) = ML j ; N (1 − θ )(1 − l ) = LP and thus N (1 − l ) = ML j + LP
(9a)
φ NK i = φ K = MK j ; (1 − φ ) NK i = (1 − φ ) K =K P and thus K = NK i = MK j + K P
(9b)
Equation (9a) asserts that the total supply of labor must be allocated either to one of the private firms
or to the public firm, while (9b) describes an analogous allocation condition for private capital.
Using these relationships, the aggregate production function for the private sector and the public
production function may be expressed as:
Y = α Y (φK) [θN (1− l)]
b
1−b
σ
(10a)
KG
J = α J [(1− φ)K] [N(1− θ)(1− l)]
d
1− d
η
KG
(10b)
The government runs a balanced budget in accordance with
τ K rK + τ w N (1 − l ) w + τ c C + T = rK P + wLP = r (1 − φ ) K + wN (1 − θ )(1 − l )
9
(11)
That is, the government finances its production costs, given by the right hand side of (11), from the
aggregate tax revenues earned on capital income, labor income, consumption, or lump-sum taxes,
where C ≡ NCi denotes aggregate consumption.6
We assume that the government ties its current rate of investment to aggregate private output
in accordance with
J = gY
(12)
where g is a chosen policy parameter. We should emphasize that (12) represents a specification of
government policy, not any kind of resource constraint on the government, which is specified by
(11). It represents the notion that the government wishes to maintain a balance between the rate of
public investment and the size of the economy.
Using (2a) and (2b), together with the full employment conditions (9a) and (9b), we may
express the government’s flow budget constraint as
b
1− b
b
1− b
C
T =  (1 − φ ) +
−τ c  Y
(1 − θ ) − τ k − τ w
θ
φ
θ
Y
φ
(13a)
so that T represents the amount of lump-sum taxation (or transfers) necessary to finance the primary
deficit and is therefore a measure of current fiscal imbalance. Defining
t
∞
∫ − s (u )(1−τ k ) du
V = ∫ T (t )e 0
0
t
b
1− b
b
1− b
C  ∫ − s (u )(1−τ k ) du
(1 − θ ) − τ k − τ w
dt = ∫  (1 − φ ) +
− τ c Ye 0
dt (13b))
φ
θ
φ
θ
Y
0
∞
where s(1 − τ k ) = r (1 − τ k ) − δ k is the implied equilibrium of interest, V measures the present
discounted value of the lump-sum taxes or transfers necessary to balance the government budget
over time, and thus is a measure of the intertemporal imbalance of the government’s budget.
Aggregating (1c) over the N agents, and recalling (4) and (6), implies market clearing in the
private goods market
K& = Y − C − δ K K
6
All private agents in the economy, whether they be employed by the private or public firms, pay taxes. The model
satisfies the conditions for Ricardian equivalence, so the abstraction of government bonds involves no loss of generality.
10
Specifically, this equation asserts that private output can be costlessly transformed into productive
private capital or consumption, and the growth rate of private capital may be written as
K&  C  Y
= 1 −  − δ K
K  Y K
(14a)
Likewise, substituting (7b) into (5), the growth rate of public capital may be written as:
K& G
Y
=g
− δG .
KG
KG
(14b)
Note also the fact that the government is cost - minimizing implies
rK P + wLP = λ dJ + λ (1 − d ) J = λ J = λ gY
(15)
Thus, the cost of producing the public capital exceeds the value of production if and only if λ > 1.
In this case the government is an inefficient producer and is in effect being subsidized by the private
sector. In the case that the public firm is maximizing profit, λ = 1 and its expenditures reduce to the
conventional expression gY .
3.
Equilibrium Dynamics
Our objective is to analyze the dynamics of the aggregate economy about a stationary growth
path. To derive the balanced growth we first impose the condition that along a balanced growth path
the output-private capital is assumed to be constant, so that private output and private grow at the
same constant rate, while the allocation of labor remains constant. We then note further that the
policy specification (12) imposes the further condition that public capital must also grow at the same
constant rate. Taking percentage changes of the aggregate production functions, (10a) and (10b), the
long run (common) equilibrium growth of output, private and public capital is:
ψ=
1− b
n
1− b − σ
(16a)
where the productive elasticities in the two sectors must satisfy the restriction:
11
σ
η
=
≡ϕ
1− b 1− d
(16b)
Given the arbitrary returns to scale in the two sectors, condition (16b) is necessary in order for the
growth rates in the two sectors to be consistent with maintaining a long-run constant ratio of public
capital to output. Using (16b) we can write the two production functions in the form
Y = α Y (φ K )b [θ N (1 − l ) K Gϕ ]1−b
(10a’)
J = α J [(1 − φ ) K ]d [ N (1 − θ )(1 − l ) K Gϕ ]1− d
(10b’)
Expressed in this way we can interpret the restriction (16b) as asserting that the externality
associated with public capital operates as an increase in the efficiency of the labor input. This is a
weak condition that we shall henceforth impose.
3.1
Transitional Dynamics
To analyze the transitional dynamics of the economy about its balanced growth path, we
express the system in terms of the stationary variables: (i) the fraction of time devoted to leisure, l
and (ii) the scale-adjusted per capita quantities:7
k=
K
N
( (1−b )
(1− b −σ )
; kG =
KG
N
( (1− b )
(1−b −σ )
; y=
Y
N
( (1−b )
(1−b −σ )
; j=
J
N
( (1−b )
(1−b −σ )
;c =
C
N
( (1− b )
(1−b −σ )
Using this notation, the scale-adjusted private and respectively public output can be written as:
y = α Y (φ k )b [θ (1 − l )]1−b kGσ
(17a)
j = α J [(1 − φ )k ]d [(1 − θ )(1 − l )]1− d kGη
(17b)
The optimality conditions then enable the dynamics to be expressed in terms of these scale-adjusted
variables as follows. First by using the no arbitrage condition which implies the equality of returns
to capital and labor in the private and public sector (2a), (2b), (5a) and (5b) and the labor and private
capital allocations (6a) and (6b), together with (11) we can derive that:
7
Under constant returns to scale, these scale adjusted per capita quantities reduce to the usual per capita quantities.
12
φ
b θ
d
=
1− b 1−θ 1− d 1−φ
(18a)
Then, substituting (17a) and (17b) into the policy relationship (12) yields:
(1 − φ ) d (1 − θ )1− d
αJ
= gαY k b − d kGσ −η (1 − l ) d −b
b
1− b
φ
θ
(18b)
Equations (18a) and (18b) provide two equations that determine the instantaneous allocation
of labor and private capital across the private and public production sectors. These solutions may be
written as:
θ = θ ( k , kG , l ; g )
(19a)
φ = φ (k , kG , l ; g )
(19b)
where8
sgn ( ∂θ ∂g ) < 0, sgn(∂θ ∂ k ) = − sgn(∂θ ∂ kG ) = sgn(∂θ ∂ l ) = sgn(d − b)
sgn ( ∂φ ∂g ) < 0, sgn(∂φ ∂ k ) = − sgn(∂φ ∂ kG ) = sgn(∂φ ∂ l ) = sgn(d − b)
These conditions have straightforward intuitive interpretations.
An increase in the rate of
government investment, g, given the current stock of productive factors requires that more labor and
private capital be allocated to the public sector. The conditions also assert that an increase in the
supply of any of the three factors of production raises the productivity of both sectors in proportion
to an amount that depends upon the respective productive elasticity. To compensate for the relative
change in productivity, and thus preserve the required balance between the rate of public investment
and private output as specified by the policy rule (12), resources must move toward the sector in
which the input has the smaller production elasticity (is less productive). It is evident from (18a)
that over time the two private factors move together across sectors ( sgn(θ&) = sgn(φ&) enabling us to
restrict our attention to say, labor.
Note that the restriction (16b) implies sgn(σ − η ) = sgn(d − b) . Explicit expressions for most of the partial derivatives
are provided in the Appendix.
8
13
The equilibrium dynamics of the economy can be expressed in terms of the stationary
variables by the following system
k& y c
= − − δ K −ψ
k k k
(20a)
k&G
y
=g
− δ G −ψ
kG
kG
(20b)
l&
y
 c&

ργ + (γ − 1)  − n + ψ  = β − (1 − τ K )b + n + δ K ;
l
φk
c

(20c)
together with
c  1 −τ w   l   1 − b 
=



y  1+τ c   1− l   ρ 
(20d)
the efficient factor allocation conditions, φ (.) and θ (.) described by (19a), and (19b), and the
production function for private output, (17a).
Equations (20a) and (20b) are the accumulation equation for scale-adjusted private and
public capital, respectively.
They are obtained by taking the time derivatives of k and kG ,
combining (14a) with (16a), and (14b) with (16a). Equation (20c) is obtained by taking the time
derivative of the optimality condition (8a) and combining it with (8c), noting that the equilibrium
rate of return on capital r = by φ k . Equation (20d) is obtained by dividing (8b) by (8a), and noting
that the equilibrium real wage is w = (1 − b)Y Nθ (1 − l ) . It implies that the consumption to output
ratio, given leisure, is increasing with a decrease in the tax on labor income and tax on consumption.
In the Appendix we show how this system can be reduced to an autonomous set of
differential equations in k , kG , and l , which then form the basis for our subsequent numerical work.
This third order system has two sluggish variables, k and kG , and one jump variable, l. To yield a
well behaved dynamic behavior we require that the eigenvalues of this system consist of two stable
and one unstable root, a property that we found to prevail over all of our wide-ranging simulations.
3.2
Steady State
14
The steady state to this economy, denoted by “~”, is obtained by setting k& = k&G = l& = 0 in
(20a) – (20c) and can be summarized by:
 c%  y%
1 − %  % = (δ K +ψ )
 yk
(21a)
%j
= (δ G +ψ )
k%
(21b)
G
(1 − τ K )
y%
= β + δ K − (γ − 1)ψ + nγ
%
φ k%
(21c)
together with the production functions, (17a) and (17b), the sectoral allocations (18a, 18b), and the
optimal consumption-leisure condition (20d). Equation (21a) describes the growth of private output,
given consumption, necessary to provide the private capital to equip the growing labor force and
replace depreciation. Equation (21b) is an analogous condition for public capital while equation
(21c) equates the long-run net return to private capital to the rate of return on consumption. These 8
equations determine the steady-state equilibrium values for k%, k% , y% , %j , l% , c%,θ%, and φ% in terms of the
G
fiscal policy instruments and other structural parameters.
From the steady-state equilibrium conditions we can show the following qualitative long-run
responses to the fiscal shocks. By the nature of the non-scale model, the long-run growth rate is
unaffected. The responses of the scale-adjusted per capita quantities are important because they
describe the effects on the base levels on which the constant steady-state growth rate compounds.
They represent the accumulated effects on the growth rates during the transition, and as our
numerical results shall highlight, they can be substantial. These are all based on the assumption that
these are financed by lump-sum taxes in accordance with (13).
Increase in Government Investment
∂l%
<0
∂g
(22a)
15
∂k%g ∂g ∂k% ∂g ∂y% ∂g
>
>
>0
y%
k%
k%
(22b)
 ∂φ% ∂θ% 
∂θ%
∂φ%
< 0,
< 0, sgn 
−
 = sgn ( b − d ) ;
∂g
∂g
 ∂g ∂g 
(22c)
g
An increase in government investment reduces the allocation of time devoted to leisure,
increasing employment, with a greater fraction of both labor and private capital being devoted to the
public sector. The relative reduction of labor in the private sector exceeds that of capital if and only
if b > d , i.e. the private sector is relatively more capital intensive. Public investment stimulates both
public and private capital, having a greater effect on the former, thereby raising the long-run ratio of
public to private capital.
Similarly, because of the reallocation of employment it has a less
stimulating effect on private output, raising the long-run output-capital ratio.
Increase in Capital Income Tax
∂l%
>0
∂τ k
(23a)
∂k% ∂τ k ∂k%g ∂τ k ∂y% ∂τ k
<
=
<0
y%
k%
k%g
(23b)
 ∂θ% ∂φ
> % > 0, if b > d ;

∂
τ
∂τ k
k

 ∂θ% ∂φ%
=
= 0, if b = d

 ∂τ k ∂τ k
 ∂φ% ∂θ%

<
< 0, if b < d
 ∂τ k ∂τ k
(23c)
An increase in the capital income tax increases the time devoted to leisure, reducing the
overall time allocated to work. The higher tax on capital income, by impinging directly on private
investment, reduces the steady-state stock of private capital more than it does public capital or
private output, both of which decline proportionately. The effect on the sectoral factor allocation
depends critically upon the sectoral factor intensity. If b = d so that both sectors have the same
16
production function, sectoral private factor allocations remain unchanged. If b > d ( b < d ) the
decline in activity will be borne less (more) heavily by the private sector, with the capital in that
sector declining in greater proportion.
Increase in Labor Income Tax or Consumption Tax
∂l%
> 0,
∂τ i
i = w, c
(24a)
∂k% ∂τ i ∂k%g ∂τ i ∂y% ∂τ i
=
=
< 0, i = w, c
y%
k%
k%g
∂θ ∂φ
=
= 0, i = w, c
∂τ i ∂τ i
(24b)
(24c)
Both these taxes decrease labor supply, but since neither impinges directly on either form of
capital accumulation they lead to proportionate long-run changes in the two capital stocks and
output, leaving the sectoral allocations unchanged as well.
4.
Socially Optimal Government Investment
Because of the declining marginal physical product of public capital, benefits to increasing
government expenditure are limited. This is because there is socially optimal level of public capital.
To help our understanding of some of the numerical simulations we shall conduct below, it is useful
to set out the steady-state equilibrium for the centrally planned economy in which the planner
controls resources directly. The optimality conditions for such an economy comprise the production
functions (17a), (17b), the efficient allocation condition (18a), the equilibrium growth conditions
(21a), (21b), together with:
ρ
c  l 
=
  (1 − b) + q (1 − d )
y  1− l 
1− b
1− d j
=q
θ
1−θ y
j

y
(25a)
(25b)
17
b
σ y
y
j
j
+ qd − δ K =
+η
− δG
k
k
q kG
kG
(25c)
b
y
j
+ qd − δ K = β + ψ (1 − γ ) + γ n
k
k
(25d)
where q denotes the shadow price of public capital in terms of private capital. These 9 equations
determine the steady-state solutions for cˆ, yˆ , lˆ, kˆ, kˆg , ˆj ,θˆ, φˆ, and qˆ . Note that the implied optimal
investment share, ĝ is obtained from the ratio gˆ = ˆj yˆ . In contrast to the decentralized economy,
the after-tax private rates of return are replaced by the corresponding social rate of return. Thus in
equation (25a) the after-tax real wage, which appears in (20d) is replaced by the social return to a
unit of labor. Similarly, (25b) determines the socially optimal ratio of public capital to private
output. Finally, equations (25c) and (25d) equate the long-run social rates of return of investment in
the two types of capital to the rate of return on consumption.
5.
Numerical Analysis of Transitional Paths
The complexity of the model limits study of its analytical properties. Further insights into
the effects of fiscal policy can be obtained by carrying out numerical analysis of the model. We
begin by characterizing a benchmark economy, calibrating the model using parameters
representative of the US economy reported in Table 1a.
The elasticity on capital, b = 0.35 , implies that 35% of final output accrues to private capital
and the rest to labor, which grows at an annual rate of 1.5%. The elasticity s = 0.2 on public capital
implies that public capital generates a significant externality in production. This parameter lies
within the range of the consensus estimates (see Gramlich 1994).9 As a benchmark, we begin with
the case where the private and the public production functions are identical, with the elasticity on
private capital in production function for public capital d = 0.35 . However, since one of the issues
we wish to address concerns the relative sectoral capital intensities of the two production sectors, we
also consider d = 0.2, d = 0.5 , respectively.
Given b, d , σ , the elasticity of public capital
externality in the production of public capital, η is determined by (16b).
9
This is considerably less than the estimate of 0.39 suggested in Aschauer’s early work.
18
The value of γ = −1.5 implies an intertemporal elasticity of substitution in consumption of
0.4, consistent with the estimate by Ogaki and Reinhardt (1998), and well within the standard range
of estimates. The elasticity of leisure ρ = 1.75 , which accords with the standard value in the
business cycle literature, yields an equilibrium fraction of time devoted to leisure of about 0.75
which is consistent with the empirical evidence.
The benchmark tax on wage income, t w = 0.28 reflects the average marginal personal income
tax rate in the US. Given the complex nature of capital income taxes, part of which may be taxed at
a lower rate than wages and part of which at a higher rate, we have chosen the common rate, t k= 0.28
as the benchmark. The benchmark consumption tax is zero. On the expenditure side we focus on
government investment.
The choice g = 0.04 is based on US evidence, suggesting that
approximately 4% of output is allocated to government investment.
We do not introduce
government consumption explicitly, thereby implicitly assuming that it is a perfect substitute for
private consumption and produced in the private sector. The annual depreciation rates dG=0.035 and
dK=0.05 approximate the average depreciation rates for public and private capital for the US during
recent years.10
These parameters lead to the benchmark equilibria reported in Table 1b. Focusing on
d = 0.35 , the implied fraction of time devoted to leisure, l = 0.75 , accords with the empirical
evidence, as noted. Also, the total consumption-output ratio = 0.85, the private output-private
capital ratio = 0.46, and the ratio of public to private capital = 0.33, are all plausible and generally
consistent with recent empirical evidence for the United States.11 The benchmark equilibrium also
implies that over 96% of labor and private capital is employed in the final output sector, which is
also consistent with the available data.12 The table also reports the two stable eigenvalues, which for
10
Data on private and public stocks of capital have been obtained from the Table “Real Net Stock of Fixed Reproducible
Tangible Wealth for the US” in the Survey of Current Business, May 1997. Data on gross public and private investment
have been obtained from the 2002 US Economic Report of the President (see Tables B2 and B21). Using these data we
have computed the average annual depreciation rates on private and public capital over the period 1990- 1995 to be
around 4.7% and 3.4%, respectively. A depreciation rate of 5% is a common benchmark in the real business cycle
literature; see e.g. Cooley (1995). We also experimented with higher depreciation rates (double), with little effect on our
conclusions, other than to yield a somewhat higher rate of convergence, but still consistent with the empirical evidence.
11
Specifically, the private consumption-output ratio =0.67 plus the public consumption-output ratio of 0.16 yields an
overall consumption-output ratio of 0.83, the output-private capital ratio = 0.48, and the ratio of public to private capital
= 0.30. These estimates have been computed from the sources cited in footnote 10.
12
In recent years the fraction of labor employed in the government sector is around 16% With about 20% of government
expenditure being on public investment, this would suggest that around 3.2% of labor is employed in producing
19
the benchmark economy are approximately -0.031 and -0.102. these imply that the per capita output
and capital converge at the asymptotic rate of around 2.3%, consistent with much empirical
evidence.13 Finally, the steady-state growth rate, which by the non-scale nature of the economy is
independent of policy, equals 2.17%.
Table 1b also reports the equilibria when the public
investment sector is labor intensive relative to the private sector ( d = 0.2 ) and relatively capital
intensive ( d = 0.5 ). These fairly large changes in relative sectoral intensities have relatively small
effects on the equilibrium. The most interesting aspect is that if the public sector is relatively labor
intensive then in the long run it will employ 5.33% of the labor force but only 2.5% of the private
capital stock, rather than 3.85% of both. If it is relatively capital intensive then the corresponding
allocations will be 2.43% and 4.41% respectively.
Table 1c summarizes the first-best optimal allocation of resources that would be chosen by a
central planner. For the chosen production and preference parameters, we see that the long-run
equilibrium in the decentralized economy involves too much leisure and two little public investment,
relative to the optimal fraction of government of government expenditure. If both sectors have the
same technology then 89.4% of both factors should be employed in the private sector, with public
investment being approximately 12% of private output, implying a ratio of pubic to private capital of
around 65%. If the public sector is relatively capital intensive then only around 8.2% of the labor
force should be employed in that sector, while 14.2% of the capital stock should be so allocated.
The long-run rate of public investment should be significantly higher at 15.5%, with the long-run
ratio of public to private capital increased to 81.5%.
Tables 2-4 describe the consequences of various policy changes on the economy. One of our
primary concerns is the impact of the alternative policies on economic welfare. This is measured by
the optimized utility of the representative agent
∞
∞
1
W = ∫ Z (t )e − β t dt = ∫ (C / N )γ l ργ e − β t dt
γ
0
0
(26)
government capital. Data enabling us to determine the allocation of private capital between the two sectors do not
appear to be available.
13
This estimate is based on the measure of convergence proposed by Eicher and Turnovsky (1999).
20
where Z(t) denotes instantaneous utility (short-run welfare) and C/N and l are evaluated along the
equilibrium path. The welfare gains reported are calculated as the percentage change in the flow of
income necessary to maintain the level of welfare unchanged in response to the policy shock. The
short-run impact is measured by the change in Z (t ) , while thelong-run impact is summarized by the
change in the overall intertemporal index, W.
The other key measure of economic performance, intertemporal fiscal balance, has been
defined previously in equation (13b). Evaluating this measure along the balanced growth path, we
find that in the benchmark case, V=-0.947, which implies a substantial intertemporal fiscal surplus.14
When the public production function is more labor intensive the fiscal surplus is somewhat smaller
(V=-0.926), whereas when it is more capital intensive the fiscal surplus is higher (V=-0.974) due to
the fact that the leisure is lower and the level of capital and consumption are higher, which provide
higher tax revenues. In interpreting these figures, two things should be borne in mind. First,
government consumption expenditure, which is around 16% of current income, is lumped in with
private consumption, while second, our measure of T, and therefore V, is net of government
transfers, which historically for the United States have been around 12% of current income. Once
we account for these, the surplus is transformed to an intertemporal deficit.
15
In any event, the
results are insensitive to the arbitrarily chosen level of V; what is relevant is the change in V.16
5.1
Uncompensated Normalized Fiscal Changes
Table 2 describes various basic policy changes from the benchmark economy, with the
corresponding dynamic transition paths being illustrated in Figs. 1 and 2. These are uncompensated,
meaning that they lead to changes in the government’s fiscal deficit. To make valid comparisons
between alternative fiscal policies the shocks must be standardized for revenue changes and this can
14
Since both V and Y are proportional to α , set arbitrarily to 1, we interpret V as being measured in units of income.
With the base income tax rate in our model being 28%, while government investment is 4% of income, this obviously
implies a large government surplus. Adding in government consumption expenditure of 16% plus transfers of 12%
augments government outlays to around 32% of current income, which now implies a deficit.
16
We have also estimate the value of ? which if higher than 1 shows that the cost of producing in the public sector
exceeds the value of the output being produced and the results point out that if the public sector production function is
more capital intensive ?=0.806 and if it is more labor intensive ?= 1.121. Therefore, if the public production function is
more labor intensive, the government is an inefficient producer and is in effect being subsidized by the private sector.
15
21
be done in at least two natural ways, each of which provides some insight. First, they may be
constrained to lead to the same change in the government’s current deficit T. This view may be
found in the rhetoric of politicians, why in professing the need to balance the budget typically have
only a short-term time horizon in mind.17 Alternatively, we may impose the constraint that they
involve the same change in the government’s intertemporal deficit, V.18
It turns out that the
comparisons under these two forms of normalization yield slight differences. We shall focus on the
latter, but note some issues that arise when we normalize with respect to the short-term balance.
One further consideration is that as we compare economies having different production
characteristics, they also begin with different initial intertemporal deficits (see Table 1b) and this too
needs to be taken into account when imposing the intertemporal normalization. The benchmark we
consider is one where the private and public sectors have the same production function,
parameterized by b = d = 0.35 , given in Panel B of Table 2. This approximates the conventional
one-sector production technology. 19
5.1.1
Increase in Government Investment
The base case we consider is one in which the government doubles its rate of investment
from 4% to 8%, financed by lump-sum taxation. This has the effect of increasing the intertemporal
budget deficit by 15.92% and this quantity change determines the normalization we impose upon all
the other policy changes.
Not surprisingly a doubling of the rate of government investment leads to a dramatic longrun structural change in the economy. Steady-state private capital increases by 40.5%, output
increases by 35.3%, while public capital increases by over 170%, raising the long-run ratio of public
to private capital by 90%. The increase in productivity induces a reduction in leisure of 0.84
percentage points and the dramatic increase in the public investment sector induces a 3.56
percentage point reallocation of both labor and private capital to that sector.
17
Turnovsky (2003) analyzes various fiscal shocks normalizing them with respect to their effect on the short-run deficit.
This normalization is adopted by Devereux and Love (1994) in their analysis of the two-sector Lucas model and by
Turnovsky and Chatterjee (2002).
19
In fact the results we obtain in this case area generally similar to those obtained by Turnovsky (2003) using an
aggregate one-sector model.
18
22
The immediate effect of the lump-sum tax financed increase in government investment
expenditure is to reduce the private agent’s wealth, inducing him to supply more labor, thereby
raising the marginal productivity of both types of capital and raising output. The dynamic responses
of this shock are shown in the four panels of Figure 1. The initial claim on resources by the
government crowds out private investment and consumption, so that the growth of private capital is
reduced below the growth rate of population; the scale-adjusted stock of private capital therefore
initially declines. By contrast, public capital initially grows by over 8% (see (ii)), although this then
declines steadily over time as the increase in its stock reduces the return to further public investment.
As the new public capital is put in place, its productivity raises the return to private capital, thereby
stimulating its growth rate. Over time, as more public capital is accumulated and its growth rate
declines, this effect is mitigated, and the growth rate of private capital, after initially rising, begins to
decline uniformly to its unchanged steady-state level. Fig. 1 (iii) illustrates the transitional paths of
key economic variables relative to their respective initial steady-state levels. The reduction in initial
consumption and leisure, with no public expenditure benefits, leads to an initial reduction in welfare
of 3.97% (Fig. 1 (iv)). But as capital is accumulated and output and consumption grow, so does
welfare. Asymptotically, instantaneous welfare rises to around 30% above the base level, and the
overall intertemporal welfare gain, derived from (26) is about 8.6%, in terms of steady income flow.
5.1.2
Decrease in Tax on Capital
Row 2 reduces the tax on capital income. In order to increase the intertermporal deficit by
the benchmark level of 15.92%, τ k must be reduced from 0.28 to 0.2029. This also has a dramatic
long-run effect on the economy. It increases private capital by over 22% and public capital and
output proportionately by around 10.6%, reducing both the output-private capital and public capitalprivate capital ratios by almost 10%. The increase in long-run capital raises the productivity of
labor, causing a long-run substitution toward more labor and less consumption. As long as the
production functions in the two sectors are identical ( b = d ), the reallocation of resources occurs in a
balanced way, so that the relative sizes of the two sectors remain unchanged; c.f. (23c).
The dynamics are illustrated in Fig. 2A. Upon impact, the lower capital tax stimulates the
23
accumulation of private capital, so that the growth rate of private capital immediately increases to
around 3.6%. The increase in labor, all of which occurs virtually on impact, and the increase in
private capital raises the growth rate of output, and the growth rate of public capital begins to
increase as well, so that both k and k g increase as indicated in Fig. 2A.(i). With the incentive
applying directly to private capital, k increases relative to k g during the initial stages. But as private
capital increases in relative abundance its productivity declines, inducing less investment in private
capital and therefore a decline in its growth rate. This in turn reduces the productivity of public
capital, the gradually increasing growth rate of which is reversed after around 10 periods. In the
short run the lower capital tax reduces welfare by nearly 3%. This is because the lower capital
income tax attracts resources toward investment and away from consumption, while the increase in
labor productivity from the higher capital induces a substitution toward more labor and less leisure,
both of which are welfare-reducing. However, the steady increase in consumption over time implies
that instantaneous welfare rises steadily relative to the benchmark, increasing asymptotically by
about 7%, more than offsetting the initial losses. The overall increase in intertemporal welfare, (26),
is equivalent to a 2.07% increase in base income.
5.1.3
Decrease in Tax on Wage Income and Consumption
Rows 3 and 4 consider a decrease in the wage tax from 0.28 to 0.2175, and the introduction
of a consumption subsidy of -0.0488, respectively, both of which decrease V by the same fixed
amount of 15.92%. These taxes both impinge on the consumption-leisure margin and are thus
qualitatively similar. We therefore illustrate only τ w . In contrast to either g or τ k these taxes do not
impinge directly on either form of capital and consequently the time paths for the two capital stocks,
as illustrated by the phase diagram Fig. 2B(i), indicates a much more balanced accumulation path.
The contrasts in both these taxes with the capital income tax are quite striking. First, since
both τ w and τ c fall more directly on consumers, their cuts have a significantly greater positive
impact on labor supply. Second, because of their more balanced impacts both tax cuts lead to
proportionate long-run increases in public capital, private capital, and output, these effects being
substantially smaller than for the capital income tax cut. They also have effect on the long-run
24
allocations of private capital and labor between the two sectors, although they do have some
temporary impacts, we shall discuss below. Moreover, whereas the capital income tax cut leads to
short-run losses followed by large long-run gains, wage and consumption taxes lead to steadily
increasing welfare gains.
The final column yields an interesting ranking among these four fiscal policies each of which
increases the long-run deficit by the same proportionate amount. First, increasing government
investment is superior by a substantial margin. This is hardly surprising, given that we are starting
an initial government investment rate of around 4% (the recent US figure), whereas the optimal that
would be provided in the first-best equilibrium is nearly 12%. Cutting the tax on labor income
would be the next best, raising welfare by 2.26%, which is superior to cutting the tax on capital,
raising welfare by 2.07%, while introducing the consumption subsidy is inferior and would raise
welfare by only 1.39%.
The interesting observation here is the superiority of cutting the wage income tax over
cutting the capital income tax. This would appear to run counter to conventional wisdom that has
argued that capital income taxes are the most harmful.20 Here, the normalization of the tax revenues
in terms of the intertemporal deficit matters. From Table 2 we see that whereas reducing τ k to
0.2029 or τ w to 0.2175 both increase the intertemporal deficit by 15.92%, the cut in the capital tax
has a much smaller effect on the current deficit than does the wage income tax, increasing it by 9.8%
rather than 13.3%. This reflects the fact that the capital income tax has a much greater effect on
growth and thus on the long run.
If, instead, we were to normalize all changes so as to increase the short-run deficit by 13.48%
(the effect of increasing g from 0.04 to 0.08), then we would find that while the tax on labor income
could be reduced only marginally more to 0.2167, the tax on capital income could be reduced
substantially to 0.1747. These policies would lead to intertemporal welfare improvements of 2.29%
and 2.76%, respectively, so that now cutting the tax on capital would be superior to cutting the tax
on labor. But, at the same time this would be associated with a larger increase in the intertemporal
20
Chamley (1986) originally showed that in the long-run the optimal tax on capital is zero. This result depends upon
the absence of externalities and modifications of this proposition in the presence of externalities and growth has been
addressed extensively in the recent growth literature.
25
deficit, suggesting a tradeoff between the welfare of the representative agent and the intertemporal
solvency of the government.
5.1.4
The Role of Sectoral Factor Intensities
Panels A and C of Table 2 report the long-run responses to the policies we have been
considering in the case where the public production function is relatively less capital intensive,
(b = 0.35 > 0.20 = d ) , and relatively more capital intensive (b = 0.35 < 0.50 = d ) , respectively.
Overall, the main qualitative effects are robust with respect to these substantial differences in
technologies. All of the long-run welfare gains increase as the relative capital intensity of the public
investment sector increases. This is because the more productive is private capital in the public
sector the greater the benefits from growth stimulating policies.
While the quantitative effects of the various tax policies are also rather insensitive to the
production technologies, that is not so in the case of direct government investment. Table 2 suggests
that government investment is generally about 30% more expansionary if the public production
function is capital intensive than if it is labor intensive. This translates to welfare gains ranging from
7.73% and 10.94% in these two cases. The most important impacts of the different production
functions for the public capital is on the allocation of the private factors of production between the
private and public sectors. Thus, whereas with identical production functions raising the public
investment rate increases the allocation of both capital and labor in the public sector by 3.56%, these
quantities are 2.46% and 4.46%, respectively if the public sector is relatively less capital intensive
and 4.82% and 2.77% if it is relatively more capital intensive.
5.2
Compensated Changes
Many policy discussions focus on compensated fiscal changes, meaning that the policy
change is accompanied by some other accommodating change so that the government deficit
remains unchanged. As we noted before, the normalization can be with respect to the current deficit
or the intertemporal deficit. We choose the latter, and Tables 3 and 4 describe a number of changes
that hold V unchanged. Again we shall focus primarily on the case where the production functions
26
are common across the two sectors, referring to Panel B.
5.2.1
Alternatively-Financed Modes of Government Production Expenditure
To provide a common benchmark, Row 1 repeats the policy of an increase in government
investment expenditure from 0.04 to 0.08, financed by lump-sum taxes. Rows 2-4 then compare this
to the cases where the public investment is financed by the three distortionary tax rates, and where in
each case the rate is set so as to maintain the present value of the fiscal deficit unchanged. The
dynamics of capital, sectoral labor allocation, and instantaneous welfare are illustrated in Fig. 3 for
the two cases of relative sectoral factor intensities.
The most interesting results are the welfare rankings. In the short run welfare declines by
about 3% for the three cases of lump-sum tax, consumption tax, and wage tax financing. This is
because the government investment stimulates employment, leading to substitution from
consumption, while providing no initial direct benefits. Over time, as the government investment
yields its productive payoff and output rises, instantaneous welfare in all cases rises steadily. As
noted before, with lump sum tax financing it converges to a higher long-run level around 30% above
that of the benchmark economy, with a present value of 8.58%, and as before this is best. With
consumption tax, capital income tax, and wage tax financing the respective improvements in
asymptotic welfare are 25.6%, 19.5%, and 21.3%, respectively, yielding the corresponding
intertemporal welfare gains of 6.68%, 5.52%, and 4.71%, respectively. This long-run ranking
reflects the rankings in 5.1 where, for example, although lowering the consumption tax was least
beneficial, raising it is also the least harmful.
The time paths for the two capital stocks exhibit interesting differences across financing
modes. In all cases government capital increases steadily. Private capital also increases in the long
run, although in the case of capital income tax-financing it is modest. It is also subject to an initial
decline during the transition, which increases in magnitude as we switch from consumption tax
financing, to wage income tax financing, and to capital income tax financing.
Comparing Figs. 3A and 3C we see that these results are robust with respect to the nature of
the sectoral production functions. The only difference is that the welfare gains are slightly higher
27
when the productivity of private capital in the public sector is higher.
Fig. 3B illustrates the time paths for the sectoral allocation of labor. As noted earlier, the
sectoral allocation of capital follows a similar pattern and is not illustrated. We shall focus on the
case b > d , in which case the initial allocation is θ = 94.77 In the case of lump-sum tax financing,
wage income, or the consumption tax, the long-run effect of the increase in government investment
is to reduce θ by 4.71 percentage points, so that it converges to θ = 90.06 . In each case, the
reallocation instantaneously adjusts to within 0.1 percentage points of the new equilibrium. During
the initial phase of the transition a small fraction of the labor is reallocated back to the private sector,
before converging back to the new equilibrium.
In the case of capital income tax financing,
employment in the private sector is reduced by 4.46 percentage points, so that it converges to
θ = 90.31 . On impact, the labor allocation overshoots by about 0.20 percentage points and it then
overshoots again slightly during the subsequent transition. The case where the public sector is
relatively capital intensive is illustrated in Fig. 3B and is seen to be a mirror image.
5.2.2
Tax Substitution
Table 4 summarizes two forms of tax substitution. Row 1 introduces a 6% consumption tax,
which permits income taxes to be reduced to around 24% (depending upon the factor intensity) to
maintain the initial intertemporal deficit, V, unchanged. This change in the tax structure raises longrun output and public capital by around 5% and private capital by 10.8%, relative to the benchmark
economy. The shift from the wage tax to the consumption tax causes a slight reduction in long-run
leisure, with a proportionately slightly larger reduction in the consumption-output ratio.
The dynamics are illustrated in Fig. 2B. The reduction in the tax on income favors private
capital, the growth rate of which rises to nearly 2.9% on impact. This stimulates the productivity of
public capital, the growth rate of which also rises in the short run, though more modestly. The tax
switch causes an initial drop in consumption of 1% and an increase in labor supply of 1%, thus
leading to an initial deterioration in welfare. However, the higher output level and resulting higher
consumption throughout the transition causes current welfare to improve over time, doing so by
3.5% asymptotically, and a present value equivalent of nearly 1%.
28
Row 2 substitutes a higher capital income tax for a lower wage tax. The main point of
interest here is that the qualitative net benefits of this depend upon the relative capital intensities of
the two production functions. It will be mildly beneficial if the public production function is more
capital intensive and mildly undesirable otherwise.
6.
Conclusions
Recent research in fiscal policy has paid increasing attention to the role of public capital.
Virtually all models that introduce public capital employ a one-sector technology, thereby assuming
that public capital is produced in the private sector along with all other output by profit maximizing
firms. In some circumstances, the government may be more appropriately thought of as operating its
own production operation. Typically, government investment projects go out to contractors who
agree to produce the specified amount of government investment – determined by policy makers – as
efficiently as possible. In this paper, we have developed a two-sector “non-scale” production model
in which there are two types of firms, conventional profit-maximizing private firms, and what we
call “public firms”, whose objective is to produce a specified quantity of government investment
goods at minimum cost. Furthermore, we assume the production functions of the two sectors do not
in general coincide. Using this two-sector production set up we have characterized the equilibrium
dynamics, and employed this equilibrium to analyze a variety of fiscal disturbances. Because of the
complexity of the model our analysis has been carried out using simulations of a calibrated
economy. We have obtained many results, of which the following merit comment.
1. Despite the fact that fiscal policy in such an economy has no effect on the long-run
equilibrium growth rate, the slow rate of convergence implies that fiscal policy has a sustained
impact on growth rates for substantial periods during the transition. These accumulate to substantial
effects on the long-run equilibrium levels of crucial economic variables, including welfare.
2.
As an example of the accumulated impacts of policy, an increase in government
investment from 0.04 to 0.08 of private output raises the long-run level of private output by between
32% and 42%, and the ratio of public to private capital between 158% to over 200%, depending
upon the relative capital intensities of the two production sectors.
29
3. We have ranked alternative fiscal policies for normalized changes on the government’s
intertemporal deficit. For the calibrated economy, increasing the rate of government investment
from 4% to 8% clearly dominates from an intertemporal welfare viewpoint other equivalent policy
changes of (i) reducing the tax on capital income by around 8%, (ii) reducing the tax on labor
income by around 6.5%, and (iii) introducing a consumption subsidy of just under 5%. The margin
of superiority increases as the public production function becomes more capital intensive.
4. Interestingly, reducing the tax on labor income yields slightly higher welfare than does a
comparable reduction in the tax on capital income. However, the conventional inferiority of capital
income taxation applies if the revenues are normalized with regard to the current, rather than the
intertemporal, government deficit.
5. Our numerical simulations suggest the following welfare ranking for the different modes
of financing government investment.
Lump-sum tax financing dominates consumption tax
financing, which in turn dominates capital tax financing and finally wage tax financing.
The effects of tax policies are remarkably robust with respect to the relative capital intensities
of the two productive sectors. In contrast, the effects of government investment are much more
sensitive to this aspect. One conclusion of this is that one can continue to employ the one-sector
Ramsey model analyze tax policy in the presence of public capital without seriously jeopardizing the
analysis. However, one has to be more careful in analyzing the impact of public capital itself.
30
Appendix
In this Appendix we derive the linear approximation to the macrodynamic equilibrium
employed in our simulations. We begin with the four equilibrium equations (20a) – (20d) written as
k& = y − c − k (δ K +ψ )
(A.1a)
k&G = gy − kG (δ G +ψ )
(A.1b)
l&
y
 c&

ργ + (γ − 1)  − n + ψ  = β − (1 − τ K )b + δ K + n
l
φk
c

c  1 −τ w   l   1 − b 
=



y  1 + τ c   θ (1 − l )   ρ 
(A.1c)
(A.1d)
Taking the time differential of (A.1d) yields
c& y& θ& l&
l&
= − + +
c y θ l 1− l
(A.2)
and taking the time derivative of the private sector production function, (17a), yields
k&
y&
φ&
k&
θ&
l&
= b + b + (1 − b) − (1 − b)
+σ G
y
φ
k
θ
1− l
kG
Substituting into (A.2) implies
k&
c&
φ&
k&
θ&
l&
l&
=b +b −b +b
+σ G +
c
φ
k
θ
1− l
kG l
(A.3)
which involves the contemporaneous changes in the sectoral allocations. To determine these, we
take the time derivatives of the solutions (19a) and (19b)
θ&  ∂θ & ∂θ & ∂θ
=
k+
kG +
θ  ∂k
∂k g
∂l
1
l&
θ

and analogously for φ& . Substituting these expressions into (A.3) we may express c& in terms of
k&, k&G , l&
31
c&  b ∂φ b b ∂θ  &  b ∂φ σ b ∂θ  &  b ∂φ 1
1
b ∂θ  &
=
+ −
+ −
+ +b
−
 kG + 
k +
 l (A.4)
1 − l θ ∂l 
c  φ ∂k k θ ∂k 
 φ ∂l l
 φ ∂kG kG θ ∂kG 
Substituting (A.4) into (A.1c) yields the following relationship between l&, k&, and k&G

1
y
l& =  β − (1 − τ K )b
+ δ K + n − (γ − 1)(ψ − n) 
∆
φk

(γ − 1)  b ∂φ b b ∂θ  &  b ∂φ σ b ∂θ  & 
−
+ −
k +
+
−

 kG 
∆  φ ∂k k θ ∂k 
 φ ∂kG kG θ ∂kG  
(A.5)
where
 b ∂φ
1
1
b ∂θ 1 
∆ ≡ (γ − 1) 
+b
−
+  + ργ
l
1− l θ ∂ l l 
 φ ∂l
To derive the linearized equilibrium, we linearize (A.1a), (A.1b), and (A.5) in turn. Thus,
linearizing (A.1a) yields
 y ∂φ

y
y ∂θ ∂c
k& = b
+ b + (1 − b)
− − (δ K + ψ )  (k − k% )
k
θ ∂k ∂k
 φ ∂k

(A.6a)
 y ∂φ
 y ∂φ
y ∂θ
y
y ∂θ
y
∂c 
∂c 
%
+ b
+ (1 − b)
+σ
−
+ (1 − b)
− (1 − b)
−  (l − l% )
 ( kG − kG ) +  b
θ ∂k
kG ∂kG 
θ ∂l
1 − l ∂l 
 φ ∂l
 φ ∂kG
Denoting the 3 x 3 matrix by A ≡ (aij ) , we can identify this equation as
k& = a11 ( k − k% ) + a12 (kG − k%G ) + a13 (l − l% )
(A.7a)
Similarly, linearizing (A.1b) yields
 y ∂φ
y
y ∂θ 
%
k&G =  gb
+ gb + g (1 − b)
 (k − k )
φ
∂
k
k
θ
∂
k


 y ∂φ

y ∂θ
y
+  gb
+ g (1 − b)
+ gσ
− (δ G +ψ )  (kG − k%G )
kG
θ ∂kG
 φ ∂kG

 y ∂φ
y ∂θ
y 
+  gb
+ g (1 − b)
− g (1 − b)
(l − l% )

θ ∂l
1− l 
 φ ∂l
which we can identify as
32
(A.6b)
k&G = a21 (k − k% ) + a22 (kG − k%G ) + a23 (l − l% )
(A.7b)
Linearizing (A.5) is a little more complex, but in doing so, we evaluate the expression at the steady
state, where k& = k& = 0 . Following this procedure yields
G

y
y ∂θ 
1
∂φ y
l& = −(1 − τ K )b  (b − 1)
+ (b − 1) 2 + (1 − b)

2
∆
∂k φ k
φk
φθ k ∂k 

  
%
 
 k − k)
  
  b ∂φ b b ∂θ 
 b ∂ φ σ b ∂θ
− (γ − 1)  a11 
+ −
+
−
 + a21 
 φ ∂kG kG θ ∂kG
  φ ∂k k θ ∂k 
+

1 
∂φ y
y ∂θ
y 
+ (1 − b)
+σ
−(1 − τ K )b  (b − 1)

2
φθ k ∂kG
φ kkG 
∆ 
∂kG φ k

  b ∂φ b b ∂θ 
 b ∂φ σ b ∂θ
− (γ − 1)  a12 
+ −
+
−
 + a22 
 φ ∂kG kG θ ∂kG
  φ ∂k k θ ∂k 
  
%
 
 ( kG − kG )
  


1
y ∂θ
y
∂φ y
+ (1 − b)
− (1 − b)
−(1 − τ K )b  (b − 1)

2
φθ k ∂l
φ k (1 − l ) 
∆
∂l φ k

  b ∂φ b b ∂θ 
 b ∂φ σ b ∂θ   
− (γ − 1)  a13 
+ −
+
a
+ −

  (l − l% )
  φ ∂k k θ ∂k  23  φ ∂k g k g θ ∂k g   

  

(A.6c)
which we can identify as
k&G = a31 (k − k% ) + a32 (kG − k%G ) + a33 (l − l% )
(A.7c)
Note that these expressions all involve the partial derivatives
∂θ ∂φ ∂ θ ∂φ ∂θ ∂φ
∂c ∂c ∂c
,
,
,
,
,
as well as
,
,
∂k ∂k ∂kG ∂kG ∂l ∂l
∂k ∂kG ∂l
To evaluate these, we proceed in the following sequential manner. Taking differentials of the
sectoral allocation equations (19a) and (19b) we obtain
(θ (1 − φ ) )( (b − d ) k ) ; ∂φ = θ (1 − θ ) ∂φ
∂θ
=−
∂k
Γ
∂k φ (1 − φ ) ∂k
33
(A.8a)
(θ (1 − φ ) )( (σ − η ) kG ) ; ∂φ = θ (1 − θ ) ∂φ
∂θ
=−
∂kG
Γ
∂kG φ (1 − φ ) ∂kG
(A.8b)
(θ (1 − φ ) )( (b − d ) (1 − l ) ) ; ∂φ = θ (1 − θ ) ∂φ
∂θ
=−
∂l
Γ
∂k φ (1 − φ ) ∂k
(A.8c)


φ
1
b + 1 − φ d  + 1 − φ


(A.8d)
where
Γ≡
1
1−θ
θ


(1 − b) + 1 − θ (1 − d ) 
Having obtained these, the relevant partial derivatives of c are obtained by differentiating (A.1d) in
conjunction with (17a). This leads to the expressions
∂c 1 − τ w 1 − b l  y ∂φ
y
y ∂θ 
=
+b
−b 2
b

∂k 1 + τ c ρ 1 − l  φθ ∂k
θk
θ ∂k 
(A.9a)
y ∂θ
y
∂c 1 − τ w 1 − b l  y ∂φ
=
−b 2
+σ
 b
∂k g 1 + τ c ρ 1 − l  φθ ∂k g
θ ∂k g
θ kg




y
y
y
y 
∂c 1 − τ w 1 − b  
∂φ
∂θ
=
−b 2
+b
+
l  b


∂l 1 + τ c ρ   φθ (1 − l ) ∂l
θ (1 − l ) ∂k g
θ (1 − l )2  θ (1 − l ) 
(A.9b)
(A.9c)
Expressions (A.8) and (A.9) yield all the partial derivatives and substituting these partial
derivatives into the elements aij yields the linearized dynamics of the equilibrium system.
34
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36
Table 1a
Base Parameter Values
Preference and
population parameters
Production parameters
Private sector
Public sector
?=-1.5, ?= 1.75, ß=0.04, n=0.015,
αY = 1, b = 0.35, σ = 0.2, δ K = 0.05 ,
α J = 1, δ G = 0.035
d = 0.2, 0.35, 0.5; η = 0.246, 0.2, 0.1538
g= 0.04, t k=0.28, t w=0.28, t c=0.0
Fiscal parameters
Table 1b
Base equilbria
l
c/y
kg/k
d=0.20
75.02
0.848
d=0.35
75.25
d=0.50
75.50
y/k
?
f
?
0.332 0.471 94.77
97.50
2.167
0.846
0.328 0.464 96.15
96.15
2.167
0.845
0.326 0.462 97.57
95.59
2.167
eigenvalues PVGovt
deficit
-0.1023
-0.0305
-0.9263
-0.1018
-0.0305
-0.9468
-0.1017
-0.0305
-0.9740
Table 1c
Centrally Planned Economy
d=0.20
d=0.35
d=0.50
l
64.59
64.93
65.26
c/y
0.779
0.769
0.760
kg/k
0.580
0.652
0.815
y/k
0.325
0.311
0.298
?
86.92
89.38
91.82
f
93.47
39.38
85.82
?
2.167
2.167
2.167
g
0.101
0.119
0.155
Table 2
Uncompensated Fiscal Changes Normalized with Respect to Present Value of Fiscal Deficit V
∆l
θ
φ
-1.00
-4.46
-2.48
-0.42
-2.31
90.02
35.78
158.0
32.64
-13.94
-2.52
7.73
-0.39
-0.12
-0.06
-1.92
-9.66
-9.66
22.21
10.41
10.41
-9.72
-2.92
2.02
-1.55
0
0
0
0
0
9.11
9.11
9.11
-13.32
0.07
2.19
-0.93
0
0
0
0
0
5.45
5.45
5.45 -14.28
B. Public and private production functions identical: b = 0.35, d = 0.,35,σ = 0.2,η = 0.2 :
percentage change in present value of budget deficit =-15.92
0.08
1.35
-0.84
-3.56
-3.56
-0.70
-3.70
92.6
40.5
170.5
35.3
-13.48
-2.82
8.58
-0.37
0
0
-1.95
-9.67
-9.67
22.42
10.58
10.58
-9.80
-2.95
2.07
-1.58
0
0
0
0
0
9.37
9.37
9.37
-13.30
0.08
2.26
-0.94
0
0
0
0
0
5.55
5.55
5.55 -14.29
0.08
C. Public production function more intensive in capital: b = 0.35, d = 0.5,σ = 0.2,η = 0.1538 :
percentage change in present value of budget deficit = -15.92
1.39
-0.72
-2.77
-4.82
-0.96
-5.04
109.4
49.70
213.5
42.16
-13.44
-3.24
10.94
-0.36
0.06
0.10
-1.99
-9.75
-9.75
22.77
10.81
10.81
-9.82
-2.98
2.14
-1.63
0
0
0
0
0
9.73
9.73
9.73
-13.24
0.09
2.37
-0.96
0
0
0
0
0
5.69
5.69
5.69
-14.29
0.10
1.44
Short-run
c
 kg 
 y
∆


∆
∆




∆(T
)
∆(k)
∆(k
)
welf.
gains
∆(y)
%
%
 k
g
 y
 k
%
percent
points
%
points
%
%
%
%
%
A. Public production function less intensive in capital: b = 0.35, d = 0.20,σ = 0.2,η = 0.246 :
percentage change in present value of budget deficit = -15.92
%
points
1. Increase in g
from 0.04 to 0.0778
2. Decrease in τ k
from 0.28 to 0.2035
3. Decrease in τ w
from 0.28 to 0.2190
4. Decrease in τ c
from 0.0 to -0.0481
1. Increase in g
from 0.04 to 0.08
2. Decrease in τ k
from 0.28 to 0.2029
3. Decrease in τ w
from 0.28 to 0.2175
4. Decrease in τ c
from 0.0 to -0.0488
1. Increase in g
from 0.04 to 0.0882
2. Decrease in τ k
from 0.28 to 0.2014
3. Decrease in τ w
from 0.28 to 0.2153
4. Decrease in τ c
from 0.0 to -0.0498
Long-run
welf. gains
percent
Table 3:
Revenue Neutral Increase in government Investment from g = 0.04 to 0.08
Normalized with Respect to Present Value of Fiscal Deficit V
θ
φ
%
points
%
points
%
points
-1.05
-4.71
-2.38
-0.45
-2.44
95.1
37.5
168.4
34.2
-2.74
8.10
-0.60
-4.46
-2.25
1.68
10.3
120.6
8.33
139.0
19.5
0.48
5.12
0.94
-4.71
-2.38
-0.45
-2.44
95.1
22.6
139.3
19.6
-2.89
4.38
-0.01
-4.71
-2.38
-0.45
-2.44
95.1
29.6 153.0 26.5
-2.77
B. Public and private production functions identical: b = 0.35, d = 0.,35,σ = 0.2,η = 0.2
6.22
-0.84
-3.56
-3.56
-0.70
-3.70
92.6
40.5 170.5 35.3
-2.82
2. Financed by τ k
-3.56
-3.56
1.44
8.56
117.1
10.9 140.8
20.4
0.36
from 0.28 to 0.3614 -0.44
3. Financed by τ w
1.14
-3.56
-3.56
-0.70
-3.70
92.6
25.0 140.7 20.3
-3.01
from 0.28 to 0.3532
4. Financed by τ c
0.17
-3.56
-3.56
-0.70
-3.70
92.6
32.5 155.1 27.6
-2.86
from 0to 0.0556
C. Public production function more intensive in capital: b = 0.35, d = 0.5,σ = 0.2,η = 0.1538
8.58
∆l
Short-run
c
 kg 
 y
∆ 
∆ 
∆ 
∆(k)
∆(k
)
welf.
gains
∆(y)
 k
g
 y
 k
percent
%
%
%
%
%
%
A. Public production function less intensive in capital: b = 0.35, d = 0.20,σ = 0.2,η = 0.246
Long-run
welf. gains
percent
1. Financed by T
2. Financed by τ k
from 0.28 to 0.3623
3. Financed by τ w
from 0.28 to 0.3524
4. Financed by τ c
from 0to 0.0565
1. Financed by T
5.52
4.71
6.68
1. Financed by T
2. Financed by τ k
from 0.28 to 0.3550
3. Financed by τ w
from 0.28 to 0.3486
4. Financed by τ c
from 0to 0.0506
-0.60
-2.31
-4.03
-0.81
-4.22
91.6
42.1
172.2
36.1
-2.46
9.37
-0.25
-2.42
-4.23
1.15
6.69
113.4
14.6
144.5
22.3
0.42
6.44
1.24
-2.31
-4.03
-0.81
-4.33
91.6
27.3
144.0
21.8
-2.67
5.64
0.32
-2.31
-4.03
-0.81
-4.22
91.6
34.7
158.0
29.0
-2.52
7.57
Table 4:
Tax Substitutions:
Normalized with Respect to Present Value of Fiscal Deficit V
∆l
%
points
θ
φ
%
points
%
points
c
∆ 
 y
%
 y
∆ 
 k
%
 kg 
∆ 
 k
%
∆(k)
∆(k g )
∆(y)
%
%
%
Short-run
welf. gains
percent
Long-run
welf. gains
percent
Public production function less intensive in capital: b = 0.35, d = 0.20,σ = 0.2,η = 0.246
1. τ c = 0.06
τ k = τ w = 0.2400
2. τ w = 0.1660
τ k = 0.4, τ c = 0
-0.12
-0.06
-0.03
-1.00
-5.29
-5.29
10.80
4.93
4.93
-1.56
0.96
-2.22
0.20
0.10
3.01
20.12
20.12
-18.04
-1.55
-1.55
5.05
-0.13
Public and private production function identical: b = 0.35, d = 0.,35,σ = 0.2,η = 0.2
1. τ c = 0.06
τ k = τ w = 0.2399
2. τ w = 0.1641
τ k = 0.4, τ c = 0
-0.11
0
0
-1.02
-5.27
-5.27
10.84
5.00
5.00
-1.57
0.98
-2.22
0
0
3.04
20.0
20.0
-17.84
-1.40
-1.40
5.10
0.03
Public production function more intensive in capital: b = 0.35, d = 0.5,σ = 0.2,η = 0.1538
1. τ c = 0.06
τ k = τ w = 0.2398
2. τ w = 0.1621
τ k = 0.4, τ c = 0
-0.11
0.03
0.05
-1.02
-5.24
-5.24
10.87
5.06
5.06
-1.57
1.01
-2.34
-0.10
-0.18
3.03
19.77
19.77
-17.47
-1.13
-1.13
5.11
0.10
Figure 1: Increase in Government Investment
pub. cap.
B
0.35
0.3
0.25
priv. cap.
0.425 0.45 0.475
0.525 0.55
0.15
A
(i) phase diagram
growth rates
0.07
0.06
public capital
output
0.05
consumption
private capital
0.04
0.03
0.02
t
10
20
30
40
50
60
(ii) growth rates
relative levels
private capital
1.3
output
consumption
1.2
1.1
labor
t
20
40
60
80
100
(iii) transitional paths
welfare
1.3
1.25
1.2
1.15
1.1
1.05
t
20
40
60
(iv) welfare
80
100
Figure 2B Decrease in Wage Income Tax
Figure 2A: Decrease in Capital Income Tax
B
pub. cap.
0.146
pub. cap.
B
0.146
0.144
0.144
0.142
0.142
priv. cap.
priv. cap.
0.42
0.44
0.46
0.48
A
0.138
0.138
0.136
0.136
0.134
0.134
A
(i) phase diagram
growth rates
0.034
0.032
0.03
0.028
0.4150.420.4250.430.4350.440.445
(i) phase diagram
growth rates
private capital
private capital
0.025
consumption
public capital
output
public capital
0.024
consumption
0.026
0.024
output
0.023
0.022
t
t
10
20
30
40
50
10
60
20
30
40
50
60
(ii) growth rates
(ii) growth rates
private capital
relative levels
relative levels
1.09
private capital
output
1.2
1.08
1.15
1.07
output
1.1
labor
1.06
consumption
labor
1.05
1.05
consumption
1.04
t
t
20
40
60
80
20
100
(iii) transition paths
40
60
80
100
80
100
(iii) transition paths
welfare
welfare
1.05
1.06
1.04
1.04
1.03
1.02
1.02
t
20
40
60
80
1.01
100
t
0.98
20
(iv) welfare
40
60
(iv) welfare
Figure 3:Increase in Government Investment under Alternative Financing
A. private sector more capital intensive (b=0.35, d=0.20)
B. public sector morecapital intensive (b=0.35, d=0.50)
pub. cap.
cons. tax
0.35
cap. tax
wage tax
0.3
lump-sum
tax
pub. cap.
0.25
0.25
cons. tax
0.35
wage tax
0.3
cap. tax
lump-sum
tax
priv. cap.
0.375
priv. cap.
0.425 0.450.475 0.5 0.525
0.45
0.15
0.5
0.55
0.15
(i) phase diagram
(i) phase diagram
q
90.4
q
95.275
cap. tax
95.25
90.35
wage tax
95.225
90.3
90.25
95.15
lump-sum tax
90.15
cons. tax
95.175
wage tax
90.2
lump-sum tax
95.2
cons. tax
95.125
cap. tax
t
20
90.05
40
60
80
100
t
20
(initial θ = 94.77)
60
(initial θ = 97.57)
80
welfare
lump-sum tax
lump-sum tax
1.3
cons. tax
wage tax
cons. tax
1.25
wage tax
1.2
1.25
1.2
cap. tax
1.15
1.1
1.05
1.05
t
40
60
(iii) welfare
80
cap. tax
1.15
1.1
20
100
(ii) sectoral labor allocation
(ii) sectoral labor allocation
welfare
40
100
t
20
40
60
(iii) welfare
80
100
Figure 4: Substitution ofConsumption Tax for Uniform Income Tax
A. private sector more capital intensive (b=0.35, d=0.20)
pub.cap.
0.139
B. public sector more capital intensive (b=0.35, d=0.50)
pub.cap.
0.141
0.138
priv.cap.
0.137
0.43
0.136
0.139
0.135
0.138
0.134
0.137
0.133
0.44
0.45
0.136
priv.cap.
0.41
0.42 0.43
0.44
(i) phase diagram
0.135
(i) phase diagram
q
97.61
q
94.78
94.77
97.605
94.76
t
20
94.75
40
60
80
100
97.595
94.74
94.73
97.59
94.72
97.585
94.71
97.58
t
20
40
60
80
97.575
100
(initial θ = 97.57)
(initial θ = 94.77)
(ii) sectoral labor allocation
(ii) sectoral labor allocation
welfare
welfare
1.03
1.03
1.02
1.02
1.01
1.01
t
t
20
40
60
80
20
100
40
60
0.99
0.99
(iii) welfare
(iii) welfare
80
100